Polymer memristors are preeminent candidates for low-power edge computing paradigms. Poly[chalcogenoviologen-alt-triphenylamine] (PCVTPA) has been synthesized by direct coupling of chalcogeno-viologen as electron acceptor and 4-(bromomethyl)-N-(4-(bromo-methyl)phenyl)-N-phenylaniline as electron donor. The introduction of chalcogen atoms (S, Se, Te) into viologen scaffolds can greatly improve electrical conductive, electrochemical, and electrochromic properties of the materials when compared with the conventional viologens. Taking PTeVTPA as an example, the as-fabricated electronic device with a configuration of Al/PTeVTPA/ITO exhibits excellent multilevel storage and history-dependent memristive switching performance. Associated with the unique memristive behavior, the PTeVTPA-based device can not only be used to emulate the synaptic potentiation/depression, the human's learning and memorizing functions, and the transition from short-term synaptic plasticity to long-term plasticity but also carry out decimal arithmetic operations as well. This work will be expected to offer a train of new thought for constructing high-performance synaptic biomimicking and neuromorphic computing system in the near future.